# Expectation hypothesis, expectation under which measure?

As I understand the expectation hypothesis says that the implied forward rate, can be used to predict future spot rates?

If $$r_{0,2}$$ is the rate for a zero coupon bond maturing in two years, and the same with $$r_{0,1}$$. And $$r_{1,2}$$ is the rate fora zero coupon bond sold at time $$1$$ and maturing at time $$2$$. Do we then have

$$1+2r_{0,2}=(1+r_{0,1})(1+r_{1,2})?$$

And by taking expectation we have $$E[r_{1,2}]=\frac{1+2r_{0,2}}{1+r_{0,1}}-1?$$

From what I understand there are 3 possible ways to take the expectation, it is under the real world measure, the risk neutral measure or the forward measure. Which is used?

What exaxtly does the expectation hypothesis mean?

• The equation I usually see is $(1+r_{0,2})^2=(1+r_{0,1})(1+{r}_{1,2})$. – noob2 Sep 14 '20 at 0:53
• @noob2 It is just a matter of how you define $r_{0,2}$, you define it as the annual effective interest rate, I define it as the nominal interest rate convertible 0.5 times a year. In your case you define it as $P(0,2)^{-0.5}-1$, I define it as $0.5(P(0,2)^{-1}-1)$. The results are the same. – user394334 Sep 14 '20 at 1:49
• Whatever. In any case only $r_{0,1}$ and $r_{0,2}$ can be observed at time 0. From the above equation we can compute $r_{1,2}$. What is the meaning of this number? If the Expectations Hypothesis is true this number represents the expected value (or unbiased forecast) of the yield of a 1 period ZCB issued at time 1 and maturing at time 2. Sometmes the ZCB will have a higher yield than this, sometimes a lower, but on average it will be this. – noob2 Sep 14 '20 at 3:44
• @noob2 The question is: expectation under which measure? What measure are they talking about? – user394334 Sep 14 '20 at 6:21
• Depends on context. But most commonly this hypothesis is applied in the context of derivative pricing under the Risk Neutral Measure. Empirically the EH under the Real Measure does not seem to hold and a Term Premium exists in that case (the two year ZCB earns more on avg than the two one year ZCB's in sequence). But its up to the author of the paper to tell you what he/she is assuming. – noob2 Sep 14 '20 at 13:26